渗透压对红血球脊峰移动影响的变分法研究
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摘要
红血球是自然界比较简单的细胞之一，自从它被人们认识以来，对它的观察、研究比较多。在过去的许多物理学的理论研究中，一般把红血球作为封闭的流体膜来处理。在此基础上，1973年，Helfrich提出了著名的自发曲率弹性自由能模型。运用这一模型，近来人们解释了双面凹扁椭球特殊形状、闪烁现象、玻璃效应等红血球的一系列现象的物理机制。
1999年，欧阳等从大量的观测资料中发现，红血球双面凹扁椭球形状的脊峰在某种不同浓度的溶液中会发生位置的变化，这就是所谓的红血球的脊峰移动现象。随即，刘全慧等用微分法巧妙的在Helfrich自发曲率弹性自由能模型基础上对此进行了解释，认为这是由于渗透压和细胞膜表面弹性系数的变化引起了红血球峰值半径的改变。但他们的研究结论只适用于这两个参数的微小变化，当渗透压和弹性系数发生较大改变时，他们并不能给出相应的结论。
本文用U数曲线来描述红血球的形状，从Helfrich弹性自由能模型出发，运用变分法研究了渗透压和弹性系数在合理的大范围内变化时，红血球的脊峰移动的情况，并与实验进行了比较。发现当表面弹性系数在(-0.15～0.3)×10~(-8)N／m中取值时，随着渗透压的增加，红血球的峰值位置朝细胞中心移动，反之亦然。这一结论与实验观察资料吻合的非常好。
由于本研究是从闭合流体膜理论出发的，因此本文预测这一结论也同样适用于囊泡的情况，并进一步从理论上分析了表面弹性系数在更大范围内取值时的峰值移动的情况。
Human red blood cells (RBC) are the simplest in nature. Large quantities of observations and researches on them have been done since they were discovered. In the past theoretical physical study, the RBC was generally dealed as closed fluid membrane. Then in 1973 Helfrich brought forward the famous spontaneous curvature Frank energy model and with the model, the RBC circular biconcave special shape, flicker phenomenon and the glass rod effect etc have been explained recently.
In 1999, from many observation data Ou-Yang Zhong-Can with his copartners found that the apex of RBC could shift when they were exposed in the different concentrations of certain chemical agents namely the so-called apex shift phenomenon. Then Liu Quan-Hui et al skillfully explained it based on the Helfrich spontaneous curvature Frank energy model by differential coefficient method. They pointed out that the radius of the apex changed depending on the change of the osmotic pressure between the ambient and the internal environments and the surface tension coefficient. But there existed a problem in their theory that the two parameters only can take tiny variation corresponding to their research result. And when the two parameters vary in a long range, they haven't given out the result that we expected.
In this paper by variation method the apex shift phenomenon is studied when the osmotic pressure and the surface tension coefficient vary in a physically long range based on the Helfrich Frank energy model while using U curve to describe the RBC shape and compare last result with the experiments. It is demonstrated that when the surface tension coefficient varies between (-0.15- 0.3) 10-8N/m the radius of the apex of the biconcave shape can shift toward to the center depending on the increase of the osmotic pressure, and vice verse. It fits the experiments result very well.
Since the research bases on the fluid membrane theory, so in the paper we forecast our result can be applied to all the fluid vesicles, and some further theoretically analysis on the details of the apex shift also are made when the surface tension coefficient take
the values out of the above range.
1999年，欧阳等从大量的观测资料中发现，红血球双面凹扁椭球形状的脊峰在某种不同浓度的溶液中会发生位置的变化，这就是所谓的红血球的脊峰移动现象。随即，刘全慧等用微分法巧妙的在Helfrich自发曲率弹性自由能模型基础上对此进行了解释，认为这是由于渗透压和细胞膜表面弹性系数的变化引起了红血球峰值半径的改变。但他们的研究结论只适用于这两个参数的微小变化，当渗透压和弹性系数发生较大改变时，他们并不能给出相应的结论。
本文用U数曲线来描述红血球的形状，从Helfrich弹性自由能模型出发，运用变分法研究了渗透压和弹性系数在合理的大范围内变化时，红血球的脊峰移动的情况，并与实验进行了比较。发现当表面弹性系数在(-0.15～0.3)×10~(-8)N／m中取值时，随着渗透压的增加，红血球的峰值位置朝细胞中心移动，反之亦然。这一结论与实验观察资料吻合的非常好。
由于本研究是从闭合流体膜理论出发的，因此本文预测这一结论也同样适用于囊泡的情况，并进一步从理论上分析了表面弹性系数在更大范围内取值时的峰值移动的情况。
Human red blood cells (RBC) are the simplest in nature. Large quantities of observations and researches on them have been done since they were discovered. In the past theoretical physical study, the RBC was generally dealed as closed fluid membrane. Then in 1973 Helfrich brought forward the famous spontaneous curvature Frank energy model and with the model, the RBC circular biconcave special shape, flicker phenomenon and the glass rod effect etc have been explained recently.
In 1999, from many observation data Ou-Yang Zhong-Can with his copartners found that the apex of RBC could shift when they were exposed in the different concentrations of certain chemical agents namely the so-called apex shift phenomenon. Then Liu Quan-Hui et al skillfully explained it based on the Helfrich spontaneous curvature Frank energy model by differential coefficient method. They pointed out that the radius of the apex changed depending on the change of the osmotic pressure between the ambient and the internal environments and the surface tension coefficient. But there existed a problem in their theory that the two parameters only can take tiny variation corresponding to their research result. And when the two parameters vary in a long range, they haven't given out the result that we expected.
In this paper by variation method the apex shift phenomenon is studied when the osmotic pressure and the surface tension coefficient vary in a physically long range based on the Helfrich Frank energy model while using U curve to describe the RBC shape and compare last result with the experiments. It is demonstrated that when the surface tension coefficient varies between (-0.15- 0.3) 10-8N/m the radius of the apex of the biconcave shape can shift toward to the center depending on the increase of the osmotic pressure, and vice verse. It fits the experiments result very well.
Since the research bases on the fluid membrane theory, so in the paper we forecast our result can be applied to all the fluid vesicles, and some further theoretically analysis on the details of the apex shift also are made when the surface tension coefficient take
the values out of the above range.
引文
[1] P-G．德热纳J．巴杜著．卢定伟，唐玉立，孙大坤 译．冯端 校．软物质和硬科学(第一版)．长沙：湖南教育出版社．2000：81-85
[2] 周建军．闭合与开口生物膜泡的形状研究：中国科学院理论物理研究所博士论文．北京：中国科学院理论物理研究所，2002：3-6，11-24
[3] Xie Yu-Zhang and Ou-Yang Zhong-Can. Helfrich Theory of Biomembranes, Modem Physics Letters B, 1992, 6(15): 917-933
[4] M. Bessis. Living Blood Cells and Their Ultra structure. Spring-Vedag. Berlin. 1973
[5] Ou-Yang Zhong-Can and W. Helfrich. Instability and Deformation of a Spherical Vesicle by Pressure. Physical Review Letters. 1987, 59(21): 2486-2488
[6] Ou-Yang Zhong-can and Xie Yu-zhang. Curvature elasticity modulus and local flicker effect of red blood cells. Physical review A, 1990, 41 (6): 3381-3384
[7] M.B. Schneider, J. T. Jenkins and W. W. Webb. Thermal Fluctuations of Large Quasi-Spherical Bimolecular Phospholipid Vesicles. J. Physique. 1984, 45(9): 1457-1472
[8] P.B. Canham. The Minimum Energy of Bending as a Possible Explanation of the Biconcave Shape of the Human Red Blood Cell. J. Theor. Biol. 1970, 26: 61-81
[9] W. Helfrich. Elastic Properties of Lipid Bilayers: Theory and Possible Experiments. Z. Naturforsch. 1973, 28:693
[10] H. J. Deuling and W. Helfrich. The curvature elasticity of fluid membranes: a catalogue of vesicle shapes. Le Journal De Physique, 1976, 37(4): 1335
[11] M. Peterson. Modification of mean flow and turbulent energy by change in surface roughness under conditions of neutral stability. J. App. Phys. 1985, 57: 1739
[12] Hu Jian-Guo and Ou-Yang Zhong-Can. Shape Equations of the Axisymmetric Vesicle. Physical Review E, 1993, 47(1): 461-467
[13] Ou-Yang Zhong-can and Wolfgang. Helfrich. Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Physics Review A. 1989, 39(10): 5280-5288
[14] Wei-Mou Zheng and Ji-Xing Liu. Relations Between Shape Equations for Axisymmetric Vesicles. Commun. Theor. Phys. 1994, 21(3): 369-380
[15] Hiroyoshi. Naito, Masahiro. Okuda and Ou-Yang Zhong-Can. Counterexample to Some Shape Equations for Axisymmetric Vesicles. Physical Review E, 1994, 48(3): 2304-2307
[16] Weimou Zheng and Zhongcan Ou-Yang. Power Series Solutions for Vesicles. Commun. Theor. Phys. 1991, 15(3): 505-508
[17] Wei-Mou Zheng and Ji-Xing Liu. Helfrich Shape Equation for Axisymmetric Vesicles as a First Integral. Physical Review E. 1993, 48(4): 2856-2860
[18] 刘全慧．Helfrich模型一个解析解所给出的形状族与两个实验效应：中国科学院理论物理研究所博士论文．北京：中国科学院理论物理研究所，1999：36-63
[19] Quan-Hui Liu, Zhou Haijun, Ji-Xing Liu et al. Spheres and oblate ellipsoids from an analytical solution of the spontaneous-curvature fluid-membrane model. Physics Review E, 1999, 60(3): 1-7
[20] B. Angelov and I. M. Mladenov. On The Geometry of Red Blood Cell. in Geometry, Integrability and Quantization. Edited by G. L. Naber. Coral Press. 2000:27-45
[21] M. Bessis. Red Cell Shape physiology. Berlin: Pathology, Ultra structure. Springer. 1973
[22] Quan-Hui Liu, Jia Yan and Ou-Yang Zhang-Can. Apex Shift of A Circular Biconcave Vesicle Induced by Osmotic Pressure. Physics Letters A. 1999, 260:
162-167
[23] 梅向明，黄敬之 编，微分几何．第二版．北京：高等教育出版社，1988，38-163
[24] F. Matz. The Rectification of the Cassinian Oval by Means of Elliptic Functions. Am. Math. Monthly. 1895, 2: 221-222.
[25] K. H. de Haas, C. Blom, D. Ende, M. H. G. Duits and J. Mellema. Deformation of giant lipid bilayer vesicles in shear flow. Phys. Rev. E, 1997, 56:7132
[26] OU-YANG ZHONG CAN, S. LIU and XIE YU-ZHANG. On the Curvature Elasticity Theory of Fluid Membranes. Mol. Cryst. Liq. Cryst.. 1991, 204: 1465-1476
[27] S. L. Zhao, Q. H. Liu and Z. C. Ou-Yang. Apex Shift of Circular Biconcave Vesicles. Inter. J. of Modem Physics B. 2003, 17(26): 4661-4665
[28] J. F. Faucon, M. D. Mitov and P. Méléard et al. Bending elasticity and thermal fluctuations of lipid membranes. Theoretical and experimental requirements. J. Phys. (Paris). 1989, 50:2385
[29] P. Méléard, M. D. Mitov and J. F. Faucon et al. Bending elasticities of model membranes: influences of temperature and sterol content. Europhys. Lett. 1990, 11:355
[30] M. B. Schneider, J. T. Jenkins and W. W. Webb. Thermal Fluctuations of Large Cylindrical Phospholipid Vesicles. J. Phys. (Paris). 1984, 45:1456
[31] H. Naito, M. Okuda and Z. C. Ou-Yang. Experimental and theoretical studies for open membranes.Phys.Rev.E. 1996, 54:2816．在此文中很多参考文献涉及到表面弹性系数为零的情况
[32] 热力学统计物理．汪志诚．第二版．北京：高等教育出版社，1992，463
[33] Ou-Yang Zhong-Can and W. Helfrich. Instability and Deformation of a Spherical Vesicle by Pressure. Physical Review Letters. 1987, 59(21): 2486-2488
[34] M. Mutz and D. Bensimon. Observation of toroidal vesicles. Physical Review
A. 1991, 43(8): 4525-4527
[35] W. T. GOZDZ and G. GOMPPER. Shape transformations of two-component membranes under weak tension. Europhys. Lett.. 2001, 55(4): 587-593
[36] Gerald Lim H. W., Michael Wortis, and Ranjan Mukhopadhyay. Stomatocyte-discocyte-echinocyte sequence of the human red blood cell: Evidence for the bilayer-couple hypothesis from membrane mechanics. PNAS. 1999, 26:16766-16769
[2] 周建军．闭合与开口生物膜泡的形状研究：中国科学院理论物理研究所博士论文．北京：中国科学院理论物理研究所，2002：3-6，11-24
[3] Xie Yu-Zhang and Ou-Yang Zhong-Can. Helfrich Theory of Biomembranes, Modem Physics Letters B, 1992, 6(15): 917-933
[4] M. Bessis. Living Blood Cells and Their Ultra structure. Spring-Vedag. Berlin. 1973
[5] Ou-Yang Zhong-Can and W. Helfrich. Instability and Deformation of a Spherical Vesicle by Pressure. Physical Review Letters. 1987, 59(21): 2486-2488
[6] Ou-Yang Zhong-can and Xie Yu-zhang. Curvature elasticity modulus and local flicker effect of red blood cells. Physical review A, 1990, 41 (6): 3381-3384
[7] M.B. Schneider, J. T. Jenkins and W. W. Webb. Thermal Fluctuations of Large Quasi-Spherical Bimolecular Phospholipid Vesicles. J. Physique. 1984, 45(9): 1457-1472
[8] P.B. Canham. The Minimum Energy of Bending as a Possible Explanation of the Biconcave Shape of the Human Red Blood Cell. J. Theor. Biol. 1970, 26: 61-81
[9] W. Helfrich. Elastic Properties of Lipid Bilayers: Theory and Possible Experiments. Z. Naturforsch. 1973, 28:693
[10] H. J. Deuling and W. Helfrich. The curvature elasticity of fluid membranes: a catalogue of vesicle shapes. Le Journal De Physique, 1976, 37(4): 1335
[11] M. Peterson. Modification of mean flow and turbulent energy by change in surface roughness under conditions of neutral stability. J. App. Phys. 1985, 57: 1739
[12] Hu Jian-Guo and Ou-Yang Zhong-Can. Shape Equations of the Axisymmetric Vesicle. Physical Review E, 1993, 47(1): 461-467
[13] Ou-Yang Zhong-can and Wolfgang. Helfrich. Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Physics Review A. 1989, 39(10): 5280-5288
[14] Wei-Mou Zheng and Ji-Xing Liu. Relations Between Shape Equations for Axisymmetric Vesicles. Commun. Theor. Phys. 1994, 21(3): 369-380
[15] Hiroyoshi. Naito, Masahiro. Okuda and Ou-Yang Zhong-Can. Counterexample to Some Shape Equations for Axisymmetric Vesicles. Physical Review E, 1994, 48(3): 2304-2307
[16] Weimou Zheng and Zhongcan Ou-Yang. Power Series Solutions for Vesicles. Commun. Theor. Phys. 1991, 15(3): 505-508
[17] Wei-Mou Zheng and Ji-Xing Liu. Helfrich Shape Equation for Axisymmetric Vesicles as a First Integral. Physical Review E. 1993, 48(4): 2856-2860
[18] 刘全慧．Helfrich模型一个解析解所给出的形状族与两个实验效应：中国科学院理论物理研究所博士论文．北京：中国科学院理论物理研究所，1999：36-63
[19] Quan-Hui Liu, Zhou Haijun, Ji-Xing Liu et al. Spheres and oblate ellipsoids from an analytical solution of the spontaneous-curvature fluid-membrane model. Physics Review E, 1999, 60(3): 1-7
[20] B. Angelov and I. M. Mladenov. On The Geometry of Red Blood Cell. in Geometry, Integrability and Quantization. Edited by G. L. Naber. Coral Press. 2000:27-45
[21] M. Bessis. Red Cell Shape physiology. Berlin: Pathology, Ultra structure. Springer. 1973
[22] Quan-Hui Liu, Jia Yan and Ou-Yang Zhang-Can. Apex Shift of A Circular Biconcave Vesicle Induced by Osmotic Pressure. Physics Letters A. 1999, 260:
162-167
[23] 梅向明，黄敬之 编，微分几何．第二版．北京：高等教育出版社，1988，38-163
[24] F. Matz. The Rectification of the Cassinian Oval by Means of Elliptic Functions. Am. Math. Monthly. 1895, 2: 221-222.
[25] K. H. de Haas, C. Blom, D. Ende, M. H. G. Duits and J. Mellema. Deformation of giant lipid bilayer vesicles in shear flow. Phys. Rev. E, 1997, 56:7132
[26] OU-YANG ZHONG CAN, S. LIU and XIE YU-ZHANG. On the Curvature Elasticity Theory of Fluid Membranes. Mol. Cryst. Liq. Cryst.. 1991, 204: 1465-1476
[27] S. L. Zhao, Q. H. Liu and Z. C. Ou-Yang. Apex Shift of Circular Biconcave Vesicles. Inter. J. of Modem Physics B. 2003, 17(26): 4661-4665
[28] J. F. Faucon, M. D. Mitov and P. Méléard et al. Bending elasticity and thermal fluctuations of lipid membranes. Theoretical and experimental requirements. J. Phys. (Paris). 1989, 50:2385
[29] P. Méléard, M. D. Mitov and J. F. Faucon et al. Bending elasticities of model membranes: influences of temperature and sterol content. Europhys. Lett. 1990, 11:355
[30] M. B. Schneider, J. T. Jenkins and W. W. Webb. Thermal Fluctuations of Large Cylindrical Phospholipid Vesicles. J. Phys. (Paris). 1984, 45:1456
[31] H. Naito, M. Okuda and Z. C. Ou-Yang. Experimental and theoretical studies for open membranes.Phys.Rev.E. 1996, 54:2816．在此文中很多参考文献涉及到表面弹性系数为零的情况
[32] 热力学统计物理．汪志诚．第二版．北京：高等教育出版社，1992，463
[33] Ou-Yang Zhong-Can and W. Helfrich. Instability and Deformation of a Spherical Vesicle by Pressure. Physical Review Letters. 1987, 59(21): 2486-2488
[34] M. Mutz and D. Bensimon. Observation of toroidal vesicles. Physical Review
A. 1991, 43(8): 4525-4527
[35] W. T. GOZDZ and G. GOMPPER. Shape transformations of two-component membranes under weak tension. Europhys. Lett.. 2001, 55(4): 587-593
[36] Gerald Lim H. W., Michael Wortis, and Ranjan Mukhopadhyay. Stomatocyte-discocyte-echinocyte sequence of the human red blood cell: Evidence for the bilayer-couple hypothesis from membrane mechanics. PNAS. 1999, 26:16766-16769